Equivariant $K$-theory of regular compactifications: further developments

V. Uma

Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 417-441.

Résumé

We describe the $\widetilde G\times \widetilde G$-equivariant $K$-ring of $X$, where $\widetilde G$ is a factorial covering of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the description of $K_{\widetilde G\times\widetilde G}(X)$, we describe the ordinary $K$-ring $K(X)$ as a free module (whose rank is equal to the cardinality of the Weyl group) over the $K$-ring of a toric bundle over $G/B$ whose fibre is equal to the toric variety $\overline{T}^{+}$ associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of $K_{\widetilde G\times\widetilde G}(X)$ and $K(X)$ as algebras over $K_{\widetilde G\times\widetilde G}(\overline{G_{\operatorname{ad}}})$ and $K(\overline{G_{\operatorname{ad}}})$ respectively, where $\overline{G_{\operatorname{ad}}}$ is the wonderful compactification of the adjoint semisimple group $G_{\operatorname{ad}}$. In the case when $X$ is a regular compactification of $G_{\operatorname{ad}}$, we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over $\overline{G_{\operatorname{ad}}}$.

Détail

Keywords: equivariant $K$-theory, regular compactification, wonderful compactification, toric bundle.

Bibliographie

[1] V. Uma, “Equivariant $K$-theory of compactifications of algebraic groups”, Transform. Groups, 12:2 (2007), 371–406 | DOI | MR | Zbl

[2] E. Bifet, C. De Concini, C. Procesi, “Cohomology of regular embeddings”, Adv. Math., 82:1 (1990), 1–34 | DOI | MR | Zbl

[3] C. De Concini, C. Procesi, “Complete symmetric varieties”, Invariant theory (Montecatini, 1982), Lecture Notes in Math., 996, Springer, Berlin, 1983, 1–44 | DOI | MR | Zbl

[4] M. Brion, “The behaviour at infinity of the Bruhat decomposition”, Comment. Math. Helv., 73:1 (1998), 137–174 | DOI | MR | Zbl

[5] B. Iversen, “The geometry of algebraic groups”, Advances in Math., 20:1 (1976), 57–85 | DOI | MR | Zbl

[6] A. S. Merkurjev, “Comparison of the equivariant and the ordinary $K$-theory of algebraic varieties”, St. Petersburg Math. J., 9:4 (1998), 815–850 | MR | Zbl

[7] R. Steinberg, “On a theorem of Pittie”, Topology, 14:2 (1975), 173–177 | DOI | MR | Zbl

[8] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29, Cambridge Univ. Press, Cambridge, 1990, xii+204 pp. | MR | Zbl

[9] G. Vezzosi, A. Vistoli, “Higher algebraic K-theory for actions of diagonalizable groups”, Invent. Math., 153:1 (2003), 1–44 | DOI | MR | Zbl

[10] J. Gubeladze, “$K$-theory of affine toric varieties”, Homology Homotopy Appl., 1 (1999), 135–145 | DOI | MR | Zbl

[11] M. Brion, “Groupe de Picard et nombres caractéristiques des variétés sphériques”, Duke Math. J., 58:2 (1989), 397–424 | DOI | MR | Zbl

[12] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, Actualités Sci. Indust., 1337, Hermann, Paris, 1968, 288 pp. | MR | MR | Zbl | Zbl

[13] N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997, x+495 pp. | MR | Zbl

[14] C. De Concini, C. Procesi, “Complete symmetric varieties. II. Intersection theory”, Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., 6, North Holland, Amsterdam, 1985, 481–513 | MR | Zbl

[15] V. Uma, “Equivariant $K$-theory of flag varieties revisited and related results”, Colloq. Math., 132:2 (2013), 151–175 | DOI | MR | Zbl

[16] T. Hausel, B. Strumfels, “Toric hyperkähler varieties”, Doc. Math., 7 (2002), 495–534 | MR | Zbl

[17] W. Fulton, Introduction to toric varieties, The William H. Roever lectures in geometry, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993, xii+157 pp. | MR | Zbl

[18] D. A. Cox, J. B. Little, H. K. Schenck, Toric varieties, Grad. Stud. Math., 124, Amer. Math. Soc., Providence, RI, 2011, xxiv+841 pp. | DOI | MR | Zbl

[19] P. Sankaran, V. Uma, “Cohomology of toric bundles”, Comment. Math. Helv., 78:3 (2003), 540–554 ; “Errata”, 79:4 (2004), 840–841 | DOI | MR | Zbl | DOI | MR

[20] M. Brion, “Equivariant Chow groups for torus actions”, Transform. Groups, 2:3 (1997), 225–267 | DOI | MR | Zbl